How does matter behave at ultra-high densities?
So dense that the the atoms themselves collapse,
and nuclei are squeezed together. So dense that
a supertanker full of oil would be 1 mm3 in size.
Amazingly enough, there are places in the universe where this
actually happens: neutron stars.
Neutron stars are the remnants of ordinary stars that have
exploded spectacularly as supernovae. After the explosion,
gravity crushes the remaining matter into a super-dense
lump called a
neutron star
or compact star.

Below the surface of the neutron star, the pressure due to gravity is
so extreme that
there are no longer any atoms: everything is compressed down to a liquid
of neutrons, with a few protons and electrons as well. But for my research
I am
interested in even higher densities. As you burrow down into the
core of the neutron star, the pressure rises relentlessly. We don't know
what happens at the center, but if the density there is high enough
then the neutrons themselves will be crushed out of existence, liberating
the quarks inside. If that happens, the core will consist of a liquid
of quarks: quark matter. My research is about the
properties of quark matter, which turns out to have
remarkable similarities to the state of electrons in a metal,
including a type of superconductivity called color superconductivity.

Phase diagram for matter at extreme density/temperature

Physicists often summarize the properties of matter over a range of
densities and temperatures by drawing a
phase diagram.
Even for a commonplace substance like water, the phase diagram is
surprisingly complicated
when you take into account all the different types of ice.
But I am interested in the phases of matter under far more extreme conditions:
trillions of times denser than ice, and trillions of times hotter than
room temperature.
In this realm we don't have much experimental information to guide us.
In principle we should be able to calculate the behavior of
matter under such conditions, using the the theory that
describes the strong nuclear
force, which is the dominant interaction at ultra-high density.
Unfortunately, this theory,
Quantum ChromoDynamics (QCD)
is difficult to work with, and becomes intractable
when we introduce the chemical potential µ that is needed to
get a finite density of quarks. So we don't really know what the
phase diagram looks like, but we can make a reasonable guess:

Conjectured phase diagram of matter at extreme temperature and density:
For a very readable review of the QCD phase diagram, see
Simon Hands,
"The phase diagram of QCD"
(published in Contemp. Phys. 42, 209 (2001)).

Along the horizontal axis the temperature is zero, and
the density rises from the onset of nuclear matter through the transition to
quark matter. Compact stars are in this region of the phase diagram,
although it is not known whether their cores are dense enough
to reach the quark matter phase.
Along the vertical axis the temperature rises, taking
us through the crossover from the hadronic gas,
in which quarks are confined into neutrons and protons,
to the quark gluon plasma (QGP), in which quarks and gluons
are unconfined.
This is the region explored by high-energy heavy-ion colliders such
as the Relativistic Heavy-Ion Collider
(RHIC) at
Brookhaven National Laboratory, and the future Large Hadron
Collider
(LHC) at CERN.

The yellow-shaded region of the figure is where quark matter,
which we expect to be color superconducting, will occur.
What do we mean by "color superconducting"?
Think of it by analogy with electrons in a metal. At very low
temperatures (a few Kelvin), most metals suddenly become
superconducting: their resistance drops to zero. This happens
because when it is cold enough the electrons can pair up
forming Cooper pairs.
We expect that
quarks in quark matter will do something similar.